Multiplying by Zero

Date: 03/01/2002 at 08:46:14
From: Jess
Subject: Question about (M x 0)

I was going through your website (which is wonderful by the way!)
And I noticed some proofs that were really interesting about 
(-1)(-1) = +1 and so on. So I sat and thought about something like 
Mx0, and for the life of me I couldnt figure it out. We have been told 
since we were in 4th grade to learn it and accept it as truth, but 
now I am really curious as to how it is true. I was wondering if you 
could help me out. Thanks a lot!

-Jess

Date: 03/01/2002 at 09:12:41
From: Doctor Rick
Subject: Re: Question about (M x 0)

Hi, Jess.

I'm glad you like our site. Are you asking for a proof that any number 
times zero is zero? It's good to go back to basics such as this 
sometimes, as you suggest. When you learned the basics, you weren't as 
critical and logical as you are now, so they couldn't be presented in 
a way that would satisfy you now. But they can be, now that you're 
ready.

We start out with just the counting numbers 1, 2, 3, ... We find that 
the operations on these numbers have properties: commutative, 
associative, distributive, and so on. When we add zero to the number 
system, we want the same properties to continue to hold. Here is what 
we discover as a result:

 1 = 1 + 0       (0 is defined as additive identity)

 M*1 = M(1 + 0)  (property of equality: equals times equals are equal)
     = M*1 + M*0 (distributive property)

I won't bother to evaluate M*1 = M, because this isn't needed for the 
proof. I just subtract M*1 from each side:

 0 = M*0         (equals minus equals are equal)

That's what we wanted to prove. Does this answer your question?

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/